Up and Down or Down and Up - Valley View Community Unit ... 11 Skills Practice...Chapter 11 Skills Practice 581 11 Up and Down or Down and Up Exploring Quadratic Functions Vocabulary
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Up and Down or Down and UpExploring Quadratic Functions
Vocabulary
Write the given quadratic function in standard form. Then describe the shape of the graph and whether it has an absolute maximum or absolute minimum. Explain your reasoning.
2x2 5 x 1 4
Standard form: f(x) 5 2x2 2 x 2 4
Graph: The graph of this function is a parabola that opens up because the sign of the coefficient is positive. The graph has an absolute minimum because the a value is greater than 0.
10. Lea is designing a rectangular quilt. She has 16 feet of piping to finish the quilt around three sides.
Let x 5 the width of the quilt
The length of the quilt 5 16 2 2x
Let A 5 the area of the quilt
Area of a rectangle 5 width 3 length
A 5 w ? l
A(x) 5 x ? (16 2 2x)
5 x ? 16 2 x ? 2x
5 16x 2 2 x 2
5 22 x 2 1 16x
11. Kiana is making a rectangular vegetable garden alongside her home. She has 24 feet of fencing to enclose the garden around the three open sides.
Let x 5 the width of the garden
The length of the garden 5 24 2 2x
Let A 5 the area of the garden
Area of a rectangle 5 width 3 length
A 5 w ? l
A(x) 5 x ? (24 2 2x)
5 x ? 24 2 x ? 2x
5 24x 2 2 x 2
5 22 x 2 1 24x
12. Nelson is building a rectangular ice rink for the community park. The materials available limit the perimeter of the ice rink to at most 250 feet.
Let x 5 the width of the ice rink
The length of the ice rink 5 250 2 2x _________ 2
5 125 2 x
Let A 5 the area of the ice rink
Area of a rectangle 5 width 3 length
A 5 w ? l
A(x) 5 x ? (125 2 x)
5 x ? 125 2 x ? x
5 125x 2 x 2
5 2 x 2 1 125x
Use your graphing calculator to determine the absolute maximum of each function. Describe what the x- and y-coordinates of this point represent in terms of the problem situation.
13. A builder is designing a rectangular parking lot. He has 400 feet of fencing to enclose the parking lot around three sides. Let x 5 the width of the parking lot. Let A 5 the area of the parking lot. The function A(x) 5 22 x 2 1 400x represents the area of the parking lot as a function of the width.
The absolute maximum of the function is at (100, 20,000).
The x-coordinate of 100 represents the width in feet that produces the maximum area.
The y-coordinate of 20,000 represents the maximum area in square feet of the parking lot.
14. Joelle is enclosing a portion of her yard to make a pen for her ferrets. She has 20 feet of fencing. Let x 5 the width of the pen. Let A 5 the area of the pen. The function A(x) 5 2 x 2 1 10x represents the area of the pen as a function of the width.
The absolute maximum of the function is at (5, 25).
The x-coordinate of 5 represents the width in feet that produces the maximum area.
The y-coordinate of 25 represents the maximum area in square feet of the pen.
15. A baseball is thrown upward from a height of 5 feet with an initial velocity of 42 feet per second. Let t 5 the time in seconds after the baseball is thrown. Let h 5 the height of the baseball. The quadratic function h(t) 5 216 t 2 1 42t 1 5 represents the height of the baseball as a function of time.
The absolute maximum of the function is at about (1.31, 32.56).
The x-coordinate of 1.31 represents the time in seconds after the baseball is thrown that produces the maximum height.
The y-coordinate of 32.56 represents the maximum height in feet of the baseball.
16. Hector is standing on top of a playground set at a park. He throws a water balloon upward from a height of 12 feet with an initial velocity of 25 feet per second. Let t 5 the time in seconds after the balloon is thrown. Let h 5 the height of the balloon. The quadratic function h(t) 5 216 t 2 1 25t 1 12 represents the height of the balloon as a function of time.
The absolute maximum of the function is at about (0.78, 21.77).
The x-coordinate of 0.78 represents the time in seconds after the balloon is thrown that produces the maximum height.
The y-coordinate of 21.77 represents the maximum height in feet of the balloon.
17. Franco is building a rectangular roller-skating rink at the community park. The materials available limit the perimeter of the skating rink to at most 180 feet. Let x 5 the width of the skating rink. Let A 5 the area of the skating rink. The function A(x) 5 2 x 2 1 90x represents the area of the skating rink as a function of the width.
The absolute maximum of the function is at (45, 2025).
The x-coordinate of 45 represents the width in feet that produces the maximum area.
The y-coordinate of 2025 represents the maximum area in square feet of the skating rink.
18. A football is thrown upward from a height of 6 feet with an initial velocity of 65 feet per second. Let t 5 the time in seconds after the football is thrown. Let h 5 the height of the football. The quadratic function h(t) 5 216 t 2 1 65t 1 6 represents the height of the football as a function of time.
The absolute maximum of the function is at about (2.03, 72.02).
The x-coordinate of 2.03 represents the time in seconds after the football is thrown that produces the maximum height.
The y-coordinate of 72.02 represents the maximum height in feet of the football.
Walking the . . . Curve?Domain, Range, Zeros, and Intercepts
Vocabulary
Choose the term that best completes each sentence.
zeros vertical motion model interval open intervalclosed interval half-closed interval half-open interval
1. An interval is defined as the set of real numbers between two given numbers.
2. The x-intercepts of a graph of a quadratic function are also called the zeros of the quadratic function.
3. An open interval (a, b) describes the set of all numbers between a and b, but not including a or b.
4. A half-closed interval or half-open interval (a, b] describes the set of all numbers between a and b, including b but not including a. Or, [a, b) describes the set of all numbers between a and b, including a but not including b.
5. A quadratic equation that models the height of an object at a given time is a vertical motion model .
6. A closed interval [a, b] describes the set of all numbers between a and b, including a and b.
Graph the function that represents each problem situation. Identify the absolute maximum, zeros, and the domain and range of the function in terms of both the graph and problem situation. Round your answers to the nearest hundredth, if necessary.
1. A model rocket is launched from the ground with an initial velocity of 120 feet per second. The function g(t) 5 216 t 2 1 120t represents the height of the rocket, g(t), t seconds after it was launched.
Time (seconds)
Hei
ght
(fee
t)
28 26 24 22 0 2
80
280
2160
2240
2320
160
240
320
4 6 8x
y Absolute maximum: (3.75, 225)
Zeros: (0, 0), (7.5, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 7.5.
Range of graph: The range is all real numbers less than or equal to 225.
Range of the problem: The range is all real numbers less than or equal to 225 and greater than or equal to 0.
2. A model rocket is launched from the ground with an initial velocity of 60 feet per second. The function g(t) 5 216 t 2 1 60t represents the height of the rocket, g(t), t seconds after it was launched.
Time (seconds)
Hei
ght
(fee
t)
28 26 24 22 0 2
20
220
240
260
280
40
60
80
4 6 8x
y Absolute maximum: (1.875, 56.25)
Zeros: (0, 0), (3.75, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 3.75.
Range of graph: The range is all real numbers less than or equal to 56.25.
Range of the problem: The range is all real numbers less than or equal to 56.25 and greater than or equal to 0.
3. A baseball is thrown into the air from a height of 5 feet with an initial vertical velocity of 15 feet per second. The function g(t) 5 216 t 2 1 15t 1 5 represents the height of the baseball, g(t), t seconds after it was thrown.
Hei
ght
(fee
t)
24 23 22 21 0 1
2
22
24
26
28
4
6
8
2 3 4x
y
Time (seconds)
Absolute maximum: (0.47, 8.52)
Zeros: (20.26, 0), (1.20, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 1.20.
Range of graph: The range is all real numbers less than or equal to 8.52.
Range of the problem: The range is all real numbers less than or equal to 8.52 and greater than or equal to 0.
4. A football is thrown into the air from a height of 6 feet with an initial vertical velocity of 50 feet per second. The function g(t) 5 216 t 2 1 50t 1 6 represents the height of the football, g(t), t seconds after it was thrown.
Hei
ght
(fee
t)
24 23 22 21 0 1
10
210
220
230
240
20
30
40
2 3 4x
y
Time (seconds)
Absolute maximum: (1.56, 45.06)
Zeros: (20.12, 0), (3.24, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 3.24.
Range of graph: The range is all real numbers less than or equal to 45.06.
Range of the problem: The range is all real numbers less than or equal to 45.06 and greater than or equal to 0.
5. A tennis ball is dropped from a height of 25 feet. The initial velocity of an object that is dropped is 0 feet per second. The function g(t) 5 216 t 2 1 25 represents the height of the tennis ball, g(t), t seconds after it was dropped.
Hei
ght
(fee
t)
24 23 22 21 0 1
8
28
216
224
232
16
24
32
2 3 4x
y
Time (seconds)
Absolute maximum: (0, 25)
Zeros: (21.25, 0), (1.25, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 1.25.
Range of graph: The range is all real numbers less than or equal to 25.
Range of the problem: The range is all real numbers less than or equal to 25 and greater than or equal to 0.
6. A tennis ball is dropped from a height of 150 feet. The initial velocity of an object that is dropped is 0 feet per second. The function g(t) 5 216 t 2 1 150 represents the height of the tennis ball, g(t), t seconds after it was dropped.
Hei
ght
(fee
t)
24 23 22 21 0 1
40
240
280
2120
2160
80
120
160
2 3 4x
y
Time (seconds)
Absolute maximum: (0, 150)
Zeros: (23.06, 0), (3.06, 0)
Domain of graph: The domain is all real numbers from negative infinity to positive infinity.
Domain of the problem: The domain is all real numbers greater than or equal to 0 and less than or equal to 3.06.
Range of graph: The range is all real numbers less than or equal to 150.
Range of the problem: The range is all real numbers less than or equal to 150 and greater than or equal to 0.
Just Watch That Pumpkin Fly!Investigating the Vertex of a Quadratic Function
Vocabulary
Graph the quadratic function. Plot and label the vertex. Then draw and label the axis of symmetry. Explain how you determine each location.
h(t) 5 t2 1 2t 2 3
28 26 24 22 0 2
2
22
24(21, 24)
x 5 21
26
28
4
6
8
4 6 8x
y
The vertex is at (21, 24) because it is the lowest point on the curve. The axis of symmetry is 21 because the axis of symmetry is equal to the x-coordinate of the vertex.
Problem Set
Write a function that represents the vertical motion described in each problem situation.
1. A catapult hurls a watermelon from a height of 36 feet at an initial velocity of 82 feet per second.
h(t) 5 216 t 2 1 v0t 1 h0
h(t) 5 216 t 2 1 82t 1 36
2. A catapult hurls a cantaloupe from a height of 12 feet at an initial velocity of 47 feet per second.
3. A catapult hurls a pineapple from a height of 49 feet at an initial velocity of 110 feet per second.
h(t) 5 216 t 2 1 v0t 1 h0
h(t) 5 216 t 2 1 110t 1 49
4. A basketball is thrown from a height of 7 feet at an initial velocity of 58 feet per second.
h(t) 5 216 t 2 1 v0t 1 h0
h(t) 5 216 t 2 1 58t 1 7
5. A soccer ball is thrown from a height of 25 feet at an initial velocity of 46 feet per second.
h(t) 5 216 t 2 1 v0t 1 h0
h(t) 5 216 t 2 1 46t 1 25
6. A football is thrown from a height of 6 feet at an initial velocity of 74 feet per second.
h(t) 5 216 t 2 1 v0t 1 h0
h(t) 5 216 t 2 1 74t 1 6
Identify the vertex and the equation of the axis of symmetry for each vertical motion model.
7. A catapult hurls a grapefruit from a height of 24 feet at an initial velocity of 80 feet per second. The function h(t) 5 216 t 2 1 80t 1 24 represents the height of the grapefruit h(t) in terms of time t.
The vertex of the graph is (2.5, 124).
The axis of symmetry is x 5 2.5.
8. A catapult hurls a pumpkin from a height of 32 feet at an initial velocity of 96 feet per second. The function h(t) 5 216 t 2 1 96t 1 32 represents the height of the pumpkin h(t) in terms of time t.
The vertex of the graph is (3, 176).
The axis of symmetry is x 5 3.
9. A catapult hurls a watermelon from a height of 40 feet at an initial velocity of 64 feet per second. The function h(t) 5 216 t 2 1 64t 1 40 represents the height of the watermelon h(t) in terms of time t.
The vertex of the graph is (2, 104).
The axis of symmetry is x 5 2.
10. A baseball is thrown from a height of 6 feet at an initial velocity of 32 feet per second. The function h(t) 5 216 t 2 1 32t 1 6 represents the height of the baseball h(t) in terms of time t.
11. A softball is thrown from a height of 20 feet at an initial velocity of 48 feet per second. The function h(t) 5 216 t 2 1 48t 1 20 represents the height of the softball h(t) in terms of time t.
The vertex of the graph is (1.5, 56).
The axis of symmetry is x 5 1.5.
12. A rocket is launched from the ground at an initial velocity of 112 feet per second. The function h(t) 5 216 t 2 1 112t represents the height of the rocket h(t) in terms of time t.
The vertex of the graph is (3.5, 196).
The axis of symmetry is x 5 3.5.
Determine the axis of symmetry of each parabola.
13. The x-intercepts of a parabola are (3, 0) and (9, 0).
3 1 9 ______ 2 5 12 ___
2 5 6
The axis of symmetry is x 5 6.
14. The x-intercepts of a parabola are (23, 0) and (1, 0).
23 1 1 _______ 2 5 22 ___
2 5 21
The axis of symmetry is x 5 21.
15. The x-intercepts of a parabola are (212, 0) and (22, 0).
212 1 (22)
___________ 2 5 214 _____
2 5 27
The axis of symmetry is x 5 27.
16. Two symmetric points on a parabola are (21, 4) and (5, 4).
21 1 5 _______ 2 5 4 __
2 5 2
The axis of symmetry is x 5 2.
17. Two symmetric points on a parabola are (24, 8) and (2, 8).
The function in vertex form is f(x) 5 2(x 2 7.5)2 1 2.25.
12. f(x) 5 22x2 2 14x 2 12
The vertex is (23.5, 12.5).
The function in vertex form is f(x) 5 22(x 1 3.5)2 2 12.5.
Determine the x-intercepts of each quadratic function given in standard form. Use your graphing calculator. Rewrite the function in factored form.
13. f(x) 5 x2 1 2x 2 8
The x-intercepts are (2, 0) and (24, 0).
The function in factored form is f(x) 5 (x 2 2)(x 1 4).
14. f(x) 5 2x2 2 x 1 12
The x-intercepts are (24, 0) and (3, 0).
The function in factored form is f(x) 5 2(x 1 4)(x 2 3).
15. f(x) 5 24x2 1 12x 2 8
The x-intercepts are (1, 0) and (2, 0).
The function in factored form is f(x) 5 24(x 2 1)(x 2 2).
16. f(x) 5 2x2 1 18x 1 16
The x-intercepts are (28, 0) and (21, 0).
The function in factored form is f(x) 5 2(x 1 8)(x 1 1).
17. f(x) 5 1 __ 2 x2 2 1 __
2 x 2 3
The x-intercepts are (22, 0) and (3, 0).
The function in factored form is
f(x) 5 1 __ 2 (x 1 2)(x 2 3).
18. f(x) 5 1 __ 3 x2 2 2x
The x-intercepts are (0, 0) and (6, 0).
The function in factored form is f(x) 5
1 __ 3 x(x 2 6).
Identify the form of each quadratic function as either standard form, factored form, or vertex form. Then state all you know about the quadratic function’s key characteristics, based only on the given equation of the function.
19. f(x) 5 5(x 2 3)2 1 12
The function is in vertex form.
The parabola opens up and the vertex is (3, 12).
20. f(x) 5 2(x 2 8)(x 2 4)
The function is in factored form.
The parabola opens down and the x-intercepts are (8, 0) and (4, 0).
21. f(x) 5 23x2 1 5x
The function is in standard form.
The parabola opens down and the y-intercept is (0, 0).
22. f(x) 5 2 __ 3 (x 1 6)(x 2 1)
The function is in factored form.
The parabola opens up and the x-intercepts are (26, 0) and (1, 0).
23. f(x) 5 2(x 1 2)2 2 7
The function is in vertex form.
The parabola opens down and the vertex is (22, 27).
24. f(x) 5 2x2 2 1
The function is in standard form.
The parabola opens up, the y-intercept is (0, 21) and the vertex is (0, 21).
More Than Meets the EyeTransformations of Quadratic Functions
Vocabulary
Write a definition for each term in your own words.
1. vertical dilation
A vertical dilation of a function is a transformation in which the y-coordinate of every point on the graph of the function is multiplied by a common factor.
2. dilation factor
The dilation factor is the common factor by which each y-coordinate is multiplied when a function is transformed by a vertical dilation.
Problem Set
Describe the transformation performed on each function g(x) to result in d(x).